CSE 373: Data Structures and Algorithms Lecture 22: Graphs IV

Dijkstra's algorithm Dijkstra's algorithm: finds shortest (minimum weight) path between a particular pair of vertices in a weighted directed graph with nonnegative edge weights solves the "one vertex, shortest path" problem basic algorithm concept: create a table of information about the currently known best way to reach each vertex (distance, previous vertex) and improve it until it reaches the best solution in a graph where: vertices represent cities, edge weights represent driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm can be used to find the shortest route between one city and any other

Dijkstra pseudocode Dijkstra(v1, v2): for each vertex v: // Initialization v's distance := infinity. v's previous := none. v1's distance := 0. List := {all vertices}. while List is not empty: v := remove List vertex with minimum distance. mark v as known. for each unknown neighbor n of v: dist := v's distance + edge (v, n)'s weight. if dist is smaller than n's distance: n's distance := dist. n's previous := v. reconstruct path from v2 back to v1, following previous pointers.

Example: Initialization  Distance(source) = 0 Distance (all vertices but source) = A 2 B  4 1 3 10 C 2 D 2 E    5 8 4 6 F 1 G   Pick vertex in List with minimum distance.

Example: Update neighbors' distance 2 A 2 B 4 1 3 10 C 2 D 2 E   1 5 8 4 6 Distance(B) = 2 F 1 G Distance(D) = 1  

Example: Remove vertex with minimum distance G F B E C D 4 1 2 10 3 6 8 5  Pick vertex in List with minimum distance, i.e., D

Example: Update neighbors 2 A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 Distance(C) = 1 + 2 = 3 Distance(E) = 1 + 2 = 3 Distance(F) = 1 + 8 = 9 Distance(G) = 1 + 4 = 5 F 1 G 9 5

Example: Continued... Pick vertex in List with minimum distance (B) and update neighbors 2 A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 Note : distance(D) not updated since D is already known and distance(E) not updated since it is larger than previously computed F 1 G 9 5

Example: Continued... Pick vertex List with minimum distance (E) and update neighbors 2 A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 F 1 G No updating 9 5

Example: Continued... Pick vertex List with minimum distance (C) and update neighbors 2 A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 Distance(F) = 3 + 5 = 8 F 1 G 8 5

Example: Continued... Pick vertex List with minimum distance (G) and update neighbors 2 A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 F 1 G Previous distance 6 5 Distance(F) = min (8, 5+1) = 6

Example (end) 2 A 2 B 4 1 3 10 C 2 D 2 E 3 3 1 5 8 4 6 F 1 G 6 5 Pick vertex not in S with lowest cost (F) and update neighbors

Correctness Dijkstra’s algorithm is a greedy algorithm make choices that currently seem the best locally optimal does not always mean globally optimal Correct because maintains following two properties: for every known vertex, recorded distance is shortest distance to that vertex from source vertex for every unknown vertex v, its recorded distance is shortest path distance to v from source vertex, considering only currently known vertices and v

“Cloudy” Proof: The Idea Next shortest path from inside the known cloud Least cost node v THE KNOWN CLOUD competitor v' Source If the path to v is the next shortest path, the path to v' must be at least as long. Therefore, any path through v' to v cannot be shorter!